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Mirrors > Home > MPE Home > Th. List > Mathboxes > mexval | Structured version Visualization version GIF version |
Description: The set of expressions, which are pairs whose first element is a typecode, and whose second element is a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mexval.k | β’ πΎ = (mTCβπ) |
mexval.e | β’ πΈ = (mExβπ) |
mexval.r | β’ π = (mRExβπ) |
Ref | Expression |
---|---|
mexval | β’ πΈ = (πΎ Γ π ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mexval.e | . 2 β’ πΈ = (mExβπ) | |
2 | fveq2 6891 | . . . . . 6 β’ (π‘ = π β (mTCβπ‘) = (mTCβπ)) | |
3 | mexval.k | . . . . . 6 β’ πΎ = (mTCβπ) | |
4 | 2, 3 | eqtr4di 2790 | . . . . 5 β’ (π‘ = π β (mTCβπ‘) = πΎ) |
5 | fveq2 6891 | . . . . . 6 β’ (π‘ = π β (mRExβπ‘) = (mRExβπ)) | |
6 | mexval.r | . . . . . 6 β’ π = (mRExβπ) | |
7 | 5, 6 | eqtr4di 2790 | . . . . 5 β’ (π‘ = π β (mRExβπ‘) = π ) |
8 | 4, 7 | xpeq12d 5707 | . . . 4 β’ (π‘ = π β ((mTCβπ‘) Γ (mRExβπ‘)) = (πΎ Γ π )) |
9 | df-mex 34473 | . . . 4 β’ mEx = (π‘ β V β¦ ((mTCβπ‘) Γ (mRExβπ‘))) | |
10 | fvex 6904 | . . . . 5 β’ (mTCβπ‘) β V | |
11 | fvex 6904 | . . . . 5 β’ (mRExβπ‘) β V | |
12 | 10, 11 | xpex 7739 | . . . 4 β’ ((mTCβπ‘) Γ (mRExβπ‘)) β V |
13 | 8, 9, 12 | fvmpt3i 7003 | . . 3 β’ (π β V β (mExβπ) = (πΎ Γ π )) |
14 | xp0 6157 | . . . . 5 β’ (πΎ Γ β ) = β | |
15 | 14 | eqcomi 2741 | . . . 4 β’ β = (πΎ Γ β ) |
16 | fvprc 6883 | . . . 4 β’ (Β¬ π β V β (mExβπ) = β ) | |
17 | fvprc 6883 | . . . . . 6 β’ (Β¬ π β V β (mRExβπ) = β ) | |
18 | 6, 17 | eqtrid 2784 | . . . . 5 β’ (Β¬ π β V β π = β ) |
19 | 18 | xpeq2d 5706 | . . . 4 β’ (Β¬ π β V β (πΎ Γ π ) = (πΎ Γ β )) |
20 | 15, 16, 19 | 3eqtr4a 2798 | . . 3 β’ (Β¬ π β V β (mExβπ) = (πΎ Γ π )) |
21 | 13, 20 | pm2.61i 182 | . 2 β’ (mExβπ) = (πΎ Γ π ) |
22 | 1, 21 | eqtri 2760 | 1 β’ πΈ = (πΎ Γ π ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 = wceq 1541 β wcel 2106 Vcvv 3474 β c0 4322 Γ cxp 5674 βcfv 6543 mTCcmtc 34450 mRExcmrex 34452 mExcmex 34453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-mex 34473 |
This theorem is referenced by: mexval2 34489 msubff 34516 msubco 34517 msubff1 34542 mvhf 34544 msubvrs 34546 |
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